Notes on Paradigm Economy
Gereon Müller*
Abstract
It is shown that assuming instances of syncretism to be systematic in
the unmarked case may significantly reduce the number of possible inflection classes that can be generated on the basis of a given inventory
of markers, without recourse to specific constraints like Carstairs’s
(1987) Paradigm Economy Principle or Carstairs-McCarthy’s (1994)
No Blur Principle. If there is always one radically underspecified (i.e.,
elsewhere) marker per morphological domain, and if there is always
one unique marker that is chosen in cases of marker competition, it
turns out that there can be at most 2n−1 inflection classes for n markers, independently of the number of instantiations of the grammatical
category that the markers have to distribute over. The argument relies
on the notion of marker deactivation combinations.
1.
Introduction
In Distributed Morphology (see, e.g., Halle & Marantz (1993, 1994), Noyer
(1992), Halle (1997), Harley & Noyer (2003)), paradigms do not exist as
genuine objects that, e.g., grammatical constraints can refer to. Rather,
paradigms are viewed as epiphenomena – essentially, as empirical generalizations that need to be derived in some way. This is incompatible with a
more traditional view according to which paradigms exist as genuine entities in the grammar. A well-known constraint that requires the presence
of paradigms as entities of grammatical analysis is the Paradigm Economy
Principle proposed in Carstairs (1987). This constraint states that the number of inflection classes for a given inventory of markers in a given grammatical domain is limited by the highest number of allomorphic variation
in a paradigm cell. If the notion of paradigm is not available in a theory of
* For helpful comments and discussion, I would like to thank Lennart Bierkandt, Fabian
Heck, Andreas Opitz, and Jochen Trommer. This research was supported by a DFG grant
to the project Argument Encoding in Morphology and Syntax, as part of Research Unit
742 (‘Grammar and Processing of Verbal Arguments’).
Subanalysis of Argument Encoding in Distributed Morphology, 161-195
Gereon Müller & Jochen Trommer (eds.)
Linguistische Arbeits Berichte 84, Universität Leipzig 2006
162
Gereon Müller
inflectional morphology, a constraint like the Paradigm Economy Principle
cannot be adopted.1
Another constraint that has the effect of restricting the number of possible inflection classes that can be generated on the basis of a given set of inflection markers (for a given grammatical category) is Carstairs-McCarthy’s
(1994) No Blur Principle. This constraint demands that only one of the
allomorphs for a particular cell can fail to unambiguously identify inflection class. As formulated, the constraint again relies on the existence of
paradigms. However, Noyer’s (2005) Interclass Syncretism Constraint, a
constraint that is similar (though not identical) in its effects to the No Blur
Principle, is developed within (and ultimately derivable from more basic assumptions of) Distributed Morphology, and thus does without paradigms.
The latter two constraints have in common that they are fundamentally
incompatible with the idea that natural classes of inflection classes can
be referred to by inflection markers, via underspecification with respect to
more primitive, decomposed inflection class features, so as to account for
instances of syncretism (conceived of broadly as a homophony of markers)
that hold across inflection classes (‘trans-paradigmatic syncretism’). The
reason is that underspecification of inflection markers with respect to inflection class information will automatically give rise to markers that do
not unambiguously encode inflection class, in violation of No Blur and the
Interclass Syncretism Constraint.
Taken together, we can conclude that approaches which do without
paradigms and rely on decomposition and underspecification of inflection
class features are incompatible with both the Paradigm Economy Principle
and the No Blur Principle. Thus, there is a potential danger that such a
1 Other constraints that presuppose paradigms include Williams’s (1994) Basic Instantiated Paradigm Principle and McCarthy’s (2003) constraints in his Optimal Paradigms
theory; but see Bobaljik (2002) and Bobaljik (2003), respectively, for arguments against
these approaches. – Note incidentally that whereas certain other recent theories of inflectional morphology do envisage a concept of paradigm, it is far from clear whether
constraints on paradigms as they have been proposed (including the Paradigm Economy
Principle) can be straightforwardly adopted in these approaches. For instance, the concepts of paradigm that are employed in the Minimalist Morphology analyses developed in
Wunderlich (1996, 1997), and in Wiese’s (1999) stem-and-paradigm approach, are highly
abstract ones, involving underspecification, and it is difficult to see how Paradigm Economy could be verified on this basis alone. Similarly, the important building blocks of
Stump’s (2001) Paradigm Function Morphology are abstract realization rules relying on
underspecification of morpho-syntactic features; it is these realizations rules that other
constraints (such as the Bidirectional Referral Principle or a Metarule for symmetrical
syncretism in Stump (2001, ch. 4)) may refer to, not the final paradigm resulting from
the application of realization (and other) rules.
Notes on Paradigm Economy
163
theory of inflectional morphology is not restrictive enough, in the sense that
it fails to systematically narrow down the a priori possible set of inflection
classes over a given inventory of markers, to a number that is closer to what
can actually be observed in the world’s languages.
In view of this state of affairs, two strategies can be pursued. First, one
might argue that the question of how inflection classes can be constrained
is in fact irrelevant from a synchronic perspective, and assume that the
diachronic development of inflection classes, and in particular their status as
objects that are derived from some other grammatical elements (e.g., theme
vowels, derivational suffixes, etc.) at some point in a language’s history,
may account for the relatively small number of inflection classes (at least
compared to the logical possibilities) in any given domain of any given
language. Second, one can try to show that restrictions on the number
of possible inflection classes (based on a given marker inventory) follow
from independently motivated assumptions, and without invoking specific
constraints that explicitly impose restrictions on possible inflection classes
(like the Paradigm Economy Principle or the No Blur Principle). I adopt
the latter strategy in this paper.
The central assumption that I make use of is that there is a metaconstraint on inflectional systems according to which as many instances of
syncretism (in a given grammatical domain) as possible should be assumed
to be systematic, and traced back to a single morpho-syntactic specification.
This meta-principle can be formulated as in (1) (see Alexiadou & Müller
(2005)):
(1) Syncretism Principle:
Identity of form implies identity of function
(within a certain domain, and unless there is evidence to the contrary).
I assume (1) to be the null hypothesis for both a child acquiring a language, and a linguist investigating it. It is clear that (1) must be confined
to smaller domains of grammar (so as to ensure that, e.g., no common source
must be sought in English for the use of s as a nominal plural marker and
as a verbal third person present tense marker); and it is also clear that the
possibility of exceptions must not be excluded (there may be historical reasons after all why a domain of inflectional morphology might deviate from
what qualifies as optimal design according to (1)). Domain restrictions and
exceptions notwithstanding, the Syncretism Principle in (1) brings about
a shift of perspective from much recent work in inflectional morphology,
in that the burden of proof is not on considering a given instance of syncretism as systematic, but on considering it to be accidental. I will not try
to present empirical evidence for (1) in the present paper (it seems fair to
164
Gereon Müller
conclude, though, that the assumption that something like (1) underlies
inflectional systems in natural languages at least qualifies as a legitimate
research strategy).2 Rather, I will provide an argument to the effect that
the Syncretism Principle in (1) (in interaction with two simple and widely
accepted auxiliary assumptions, concerning presence of an elsewhere marker
and uniqueness of marker choice) significantly restricts the number of possible inflection classes all by itself, in a way that is made precise in the
formulation of what I call the Inflection Class Economy Theorem in (2):
(2) Inflection Class Economy Theorem:
Given a set of n inflection markers, there can be at most 2n−1 inflection
classes, independently of the number of grammatical categories that the
markers have to distribute over.
I will proceed as follows. In section 2, I briefly address the Paradigm Economy Principle and the No Blur Principle; I illustrate how these principles
restrict the number of inflection classes, and where they may raise problems
(in general, or with respect to assumptions made here). In section 3, then,
I first introduce the Inflection Class Economy Theorem and its underlying
assumptions; after that I illustrate (on the basis of some abstract examples)
how the Inflection Class Economy Theorem restricts the number of inflection classes; and only then do I show how the Inflection Class Economy
Theorem follows from the set of assumptions introduced before.
2.
Paradigm Economy
2.1. The Paradigm Economy Principle
Assuming as given the set of inflection markers that a language has at its
disposal in order to express some grammatical category (like, e.g., case), the
question arises how these markers can be grouped into inflection classes (or
paradigms – I will use these notions interchangeably in this paper). More
specifically, Carstairs (1987) brings up the issue of what the largest number
of inflection classes can be on the basis of a given set of inflection markers.
2 See, e.g., Georgi (2006) on the morphological system of argument encoding Kambera,
which looks optimally designed from the of view of the Syncretism Principle; and several
other papers in the present volume. However, also see Carstairs-McCarthy (2004) for an
explicit rejection of the idea that optimal design plays a role in morphology.
Notes on Paradigm Economy
165
An answer is provided by the Paradigm Economy Principle, which can be
formulated as in (3) (Carstairs (1987, 51)).
(3) The Paradigm Economy Principle:
When in a given language L more than one inflectional realization
is available for some bundle or bundles of non-lexically-determined
morpho-syntactic properties associated with some part of speech N,
the number of macroparadigms for N is no greater than the number
of distinct “rival” macroinflections available for that bundle which is
most genereously endowed with such rival realizations.
As a consequence, the the number of inflection classes (more precisely:
macro-inflection classes; see below) does not exceed the greatest number
of allomorphs for any instantiation of a grammatical category. Thus, a system of inflection classes as in the abstract example in (4) is predicted to be
impossible by the Paradigm Economy Principle (see Carstairs-McCarthy
(1998)). Here, the greatest number of allomorphic variation for any cell
is 2 (e.g., a and f are possible realizations of the instantiation of a given
grammatical category in cell 1; b and e are possible realizations of the instantiation of this grammatical category in cell 2; and so forth). Therefore,
the Paradigm Economy Principle demands that there can be at most two
inflection classes on the basis of this marker inventory; a system with four
inflection classes, as in (4), is excluded.
(4)
Cell
Cell
Cell
Cell
1
2
3
4
Class A Class B Class C Class D
a
a
f
f
b
e
e
e
c
c
h
h
d
d
d
g
To illustrate that the number of observable inflection classes is typically
vastly smaller than what would be predicted if markers could freely recombine, Carstairs (1987) looks at the system of present indefinite verb
inflection in Hungarian; see (5).
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Gereon Müller
(5) Hungarian present indefinite verb inflection
Indicative
Subjunctive
Sg 1 ok, ek, ök, om, em, öm ak, ek am em
2 (a)sz, (e)sz, ol, el, öl
Ø, ál, él
3
Ø, ik
on, en, ön, ék
Pl 1
unk, ünk
unk, ünk
2 (o)tok, (e)tek, (ö)tök
atok, etek
3
(a)nak, (e)nek
anak, enek
He notes that there could in principle be 276.480 inflection classes, assuming
complete independence of distribution of the markers over (macro-) inflection classes (the result of multiplying the numbers of exponents in all of the
cells). In actual fact, there are much fewer inflection classes. A superficial
analysis might take the inflectional patterns in (6) to be representative of
the system of inflection classes.
(6) Some Hungarian verbs
Indicative
olvasni
ülni
‘read’
‘sit’
Sg 1 olvas-ok
ül-ök
2 olvas-ol
ül-sz
3 olvas-Ø
ül-Ø
Pl 1 olvas-unk ül-ünk
2 olvas-tok
ül-tök
3 olvas-nak ül-nek
Subjunctive
Sg 1 olvas-ak
ülj-ek
2 olvas-Ø/-ál ülj-Ø/-él
3 olvas-on
ülj-en
Pl 1 olvas-unk ülj-ünk
2 olvas-atok ülj-etek
3 olvas-anak ülj-enek
enni
‘eat’
esz-em
esz-el
esz-ik
esz-unk
esz-tek
esz-nek
érteni
‘understand’
ért-ek
ért-esz
ért-Ø
ért-ünk
ért-etek
ért-enek
ı́rni
‘write’
ı́r-ok
ı́r-sz
ı́r-Ø
ı́r-unk
ı́r-tok
ı́r-nak
egy-em
egy-él
egy-ek
egy-ünk
egy-etek
egy-enek
értj-ek
értj-Ø/-él
értj-en
értj-ünk
értj-etek
értj-enek
irj-ak
ı́rj-Ø/-ál
ı́rj-on
ı́rj-unk
ı́rj-atok
ı́rj-anak
However, Carstairs (1987) argues that if one is willing to abstract away from
differences that are morpho-phonologically or phonologically predictable,
there are only two (macro-) inflection classes: the normal conjugation and
the ik conjugation (each with a back-vowel and a front-vowel version); cf.
(7). This is of course fully compatible with the requirements imposed by the
Paradigm Economy Principle.
Notes on Paradigm Economy
167
(7) Hungarian present indefinite conjugations: analysis
Indicative
Subjunctive
normal
ik
normal ik
Sg 1 ok
om
ak
am
2 ol (after sibilants) ol
Ø/ál
Ø/ál
asz (elsewhere)
3 Ø
ik
on
ék
Pl 1 unk
unk
unk
unk
2 (o)tok
(o)tok (o)tok (o)tok
3 (a)nak
(a)nak (a)nak (a)nak
It is worth noting that the Paradigm Economy Principle crucially relies on
the concept of macro-paradigm (or macro-inflection class). To see this, consider first the notion of inflection class proper. Following Aronoff (1994, 64),
we can assume that an inflection class is “a set of lexemes whose members
each select the same set of inflectional realizations”. Macro-paradigms (or
macro-inflection classes), in contrast, are defined by Carstairs (1987) as in
(8).
(8) Macro-Paradigm:
A macro-paradigm consists of:
a. any two or more similar paradigms whose inflectional differences
either can be accounted for phonologically, or else correlate consistently with differences in semantic or lexically determined syntactic
properties (like gender);
or
b. any paradigm which cannot be thus combined with other
paradigm(s).
Thus, differences between inflection classes that are independently predictable do not create different macro-paradigms. To see why this is needed
for the proper functioning of the Paradigm Economy Principle, consider the
set of inflection classes for noun inflection in German. The classification
in (9) assumes eight inflection classes. It is taken from Alexiadou & Müller
(2005); however, there is a similar taxonomy of inflection classes in Carstairs
(1986, 8).3
3 Carstairs (1986) actually has even more inflection classes, including ones with s as a
plural marker. – In (9), the letters m, f, and n stand for masculine, feminine, and neuter,
respectively.
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Gereon Müller
(9) German noun inflection
nom/sg
acc/sg
dat/sg
gen/sg
I: masc, neut
Hundm (‘dog’),
Schafn (‘sheep’)
Ø
Ø
Ø
(e)s
II: masc
Baumm (‘tree’)
Floßn (‘raft’)
Ø
Ø
Ø
(e)s
III: neut, masc
Buchn (‘book’),
Mannm (‘man’)
Ø
Ø
Ø
(e)s
IV: masc, neut
Strahlm (‘ray’)
Augen (‘eye’)
Ø
Ø
Ø
(e)s
nom/pl
acc/pl
dat/pl
gen/pl
(e)
(e)
(e)n
(e)
”(e)
”(e)
”(e)n
”(e)
”er
”er
”ern
”er
(e)n
(e)n
(e)n
(e)n
V: masc (‘weak’)
Planetm (‘planet’)
VI: fem
Ziegef (‘goat’)
VII: fem
Mausf (‘mouse’)
nom/sg
acc/sg
dat/sg
gen/sg
Ø
(e)n
(e)n
(e)n
Ø
Ø
Ø
Ø
Ø
Ø
Ø
Ø
VIII: fem
Drangsalf
(‘distress’)
Ø
Ø
Ø
Ø
nom/pl
acc/pl
dat/pl
gen/pl
(e)n
(e)n
(e)n
(e)n
(e)n
(e)n
(e)n
(e)n
”(e)
”(e)
”(e)n
”(e)
(e)
(e)
(e)n
(e)
The greatest number of allomorphic variation in (9) is 4, in nominative, accusative, and genitive plural contexts: (e), ”(e), ”er, (e)n (” signals umlaut
on the stem; ( ) signals a regular morpho-phonological alternation between
and zero exponence). If the (somewhat exceptional) plural marker s is also
included as a regular exponent, on a par with the other ones, the greatest
number of allomorphic variation would be 5 (but there would of course be
additional inflection classes to consider). It follows from this observation
that there can be at most 4 (5) macro-inflection classes, given the Paradigm
Economy Principle. A first approximation is the system in (10), which recognizes five inflection classes (based on (9), i.e., ignoring s).
w
(10)Macro-inflection classes for German noun declension
a. III (”er-plural)
b. V (so-called ‘weak masculines’)
c. IV/VI (en-plural; gen/sg s for masc/neut; gen/sg Ø for fem)
d. II/VII (”e-plural; gen/sg s for masc/neut; gen/sg Ø for fem)
e. I/VIII (e-plural; gen/sg s for masc/neut; gen/sg Ø for fem)
Where IV and VI differ, the difference is derivable by invoking gender in
addition to inflection class; and the same goes for II and VII, and I and
Notes on Paradigm Economy
169
VIII. However, this does not yet suffice: The Paradigm Economy Principle
permits four macro-inflection classes for German noun inflection, but there
are five inflection classes in (10). Hence, it seems that (10-d) and (10-e) must
be combined into a single, even larger macroclass, with umlaut accounted
for independently ((morpho-) phonologically). Indeed, Carstairs (1987, 58)
assumes that stem allomorphy (as with umlaut/non-umlaut alternations)
does not give rise to different macro-inflection classes (there is thus “a distinction between affixal and non-affixal inflection”).
As a further example of how the Paradigm Economy Principle works,
and how apparent counter-evidence can be handled by invoking the notion
of macro-paradigm, consider noun inflection in Russian. As shown in (11),
there are four basic inflection classes (class I contains masculine stems; class
II contains mainly feminine stems, but also some masculine stems; class III
(the ‘i-declension’) contains feminine stems; and class IV, which is similar
to class I, but still sufficiently different from class I to make postulation of a
separate inflection class unavoidable, contains only neuter stems). However,
all four classes must be split up into subclasses because of an animacy
effect: In the singular of class I, and in the plural of all inflection classes,
the genitive form is used in accusative contexts with animate stems, and
the nominative is used in accusative contexts with inanimate stems.4
(11)Russian noun inflection
a. Singular
Ia/Ibm IIa/IIbf,m IIIa/IIIbf IVa/IVbn
nom/sg
Ø
a
Ø
o
acc/sg
Ø/a
u
Ø
o
dat/sg
u
e
i
u
gen/sg
a
i
i
a
inst/sg
om
oj
ju
om
loc/sg
e
e
i
e
4 The relevant markers are set in italics. Interestingly enough, this animacy effect also
holds for class IV, where animate neuters are marked by the genitive exponent Ø in
accusative contexts; see Corbett & Fraser (1993), Fraser & Corbett (1994), and Krifka
(2003).
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Gereon Müller
b.
Plural
Ia/Ibm
nom/pl y
acc/pl y/ov
dat/pl am
gen/pl ov
inst/pl ami
loc/pl
ax
IIa/IIbf,m
y
y/Ø
am
Ø
ami
ax
IIIa/IIIbf
i
i/ej
jam
ej
jami
jax
IVa/IVbn
a
a/Ø
am
Ø
ami
ax
Thus, it seems that eight inflection classes must be postulated for Russian
noun inflection. However, the greatest number of allomorphic variation for
a given case is 4 (in the accusative singular). Again, the solution of this
apparent problem for the Paradigm Economy Principle is to hold independently established morpho-syntactic features responsible for the differences,
and thereby reduce the number of macro-paradigms. Thus, the variation in
accusative singular contexts (in class 1) and accusative plural context (in
all classes) correlates consistently with differences in semantic properties
(viz., animacy), and is thus predictable: This reduces the number of macroparadigms from 8 to 4. Furthermore, the differences between class I and class
IV are also predictable on the basis of independently given information, viz.,
gender: Hence, the number of macro-paradigms is further reduced from 4
to 3. The result is in accordance with the Paradigm Economy Principle.
Concluding so far, given the concept of macro-paradigm (or macroinflection class), apparent counter-examples to the Paradigm Economy Principle can be explained away. On this view, if a different inflectional pattern
can be described by invoking gender features, semantic features (like animacy), phonological features, or if it involves non-affixal inflection, it is
irrelevant for paradigm economy: Only those differences count which are
absolutely irreducible.
Still, from a more general point of view, there seems to be a potential tension between descriptive adequacy and explanatory adequacy with
the Paradigm Economy Principle: Without a concept like that of a macroparadigm, the Paradigm Economy Principle would be much too restrictive;
it would exclude many of the attested inflection patterns in languages with
inflection classes. However, assuming such a less rigid concept of macroparadigm as relevant for the constraint reduces the Paradigm Economy
Principle’s predictive power.
Notes on Paradigm Economy
171
2.2. No Blur
The No Blur Principle is proposed in Carstairs-McCarthy (1994, 742) as a
successor to the Paradigm Economy Principle; see (12).
(12)No Blur Principle:
Within any set of competing inflectional realizations for the same
paradigmatic cell, no more than one can fail to identify inflection class
unambiguously.
The underlying idea is that there is typically one elsewhere marker that
is not specified for inflection class, but no more than that (also see Noyer
(2005)). Just like the Paradigm Economy Principle, the No Blur Principle
blocks (what looks like) a constant reuse of inflectional material in various
inflection classes, and thereby constrains the number of possible inflection
classes that can be generated on the basis of a given inventory of markers.
The No Blur Principle can be illstrated on the basis of the strong feminine noun declensions in Icelandic (see Carstairs-McCarthy (1994, 740742)). As shown in (13), there are four strong feminine inflection classes.
These are sometimes given names according to the theme vowels (or lack
of theme vowels) in Old Norse: Fa (a-stem declension, with a subclass Fa ′
that we may ignore in the present context); Fi (i-stem declension); Fc1 (first
consonantal declension); and Fc2 (second consonantal declension).
(13)Strong feminine inflection classes in Icelandic
Fa
Fa′
Fi
vél (‘ma- drottning
mynd
(chine’) (‘queen’)
(‘picture’)
nom/sg vél-Ø
drottning-Ø mynd-Ø
acc/sg vél-Ø
drottning-u
mynd-Ø
dat/sg vél-Ø
drottning-u
mynd-Ø
gen/sg vél-ar
drottning-ar mynd-ar
nom/pl vél-ar
drottning-ar mynd-ir
acc/pl vél-ar
drottning-ar mynd-ir
dat/pl vél-um
drottning-um mynd-um
gen/pl vél-a
drottning-a
mynd-a
Fc1
geit
(‘goat’)
geit-Ø
geit-Ø
geit-Ø
geit-ar
geit-ur
geit-ur
geit-um
geit-a
Fc2
vı́k
(‘bay’)
vı́k-Ø
vı́k-Ø
vı́k-Ø
vı́k-ur
vı́k-ur
vı́k-ur
vı́k-um
vı́k-a
The interesting differences between these inflection classes are all confined
to genitive singular and nominative (and accusative) plural contexts; forms
with genitive singular markers and forms with nominative/plural markers
are the “leading forms” (“Kennformen”; see Wurzel (1987)). According to
the the No Blur Principle, only one of the allomorphs for a given instan-
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Gereon Müller
tiation of a grammatical category can fail to unambiguously identify an
inflection class. The issue is trivial in nominative, accusative, and dative
singular contexts (where there is only one default marker Ø, abstracting
away from u in Fa′ , which however clearly identifies a (sub)declension); and
in dative and genitive plural contexts as well (because, again, there is only
one default marker in each case, which is not inflection-class specific: um, a).
More interestingly, in genitive singular contexts, ur is used with inflection
class Fc2 (which it unambiguously identifies), and ar is the elsewhere marker
that is not restricted to any inflection class (and blocked in Fc2 contexts
only because there is a more specific marker). Similarly, ar is a highly specific nominative (and accusative) plural marker that is confined to inflection
class Fa; ir is an equally specific nominative (and accusative) plural marker
that is confined to inflection class Fi; and ur is a more general nominative
(and accusative) plural marker that does not bear any inflection class information and thus gets an elsewhere distribution. Since, in both cases, there
is only one marker that does not unambiguously identify an inflection class,
the No Blur Principle is fully respected by the data in (13).
However, closer scrutiny reveals that the system of Icelandic noun inflection may not be completley unproblematic for the No Blur Principle.
This constraint seems to make wrong predictions ifthe complete system of
Icelandic noun declension is taken in to account (see Kress (1982), Müller
(2005)). As shown in (14), there is more than one marker that fails to
unambiguously identify inflection class in both genitive singular and and
nominative plural contexts.5
(14)The complete
I
Ma
nom/sg ur
acc/sg
Ø
dat/sg
i
gen/sg
s
nom/pl ar
acc/pl
a
dat/pl um
gen/pl
a
system of inflection
II
III IV V
Na Fa(′ ) Mi Fi
Ø
Ø
ur Ø
Ø Ø (u) Ø Ø
i Ø (u) Ø Ø
s
ar
ar ar
Ø
ar
ir ir
Ø
ar
i ir
um um um um
a
a
a a
classes in Icelandic noun inflection
VI VII VIII IX X XI XII
Mu Mc Fc1 Fc2 Mw Nw Fw
ur ur Ø
Ø
i
a
a
Ø Ø Ø
Ø
a
a
u
i
i
Ø
Ø
a
a
u
ar ar ar ur
a
a
u
ir ur ur ur ar
u
ur
i ur ur ur
a
u
ur
um um um um um um um
a
a
a
a
a (n)a (n)a
5 A remark on notation: Ma, Mi, and Mc are the strong masculine declensions; Na is
the strong neuter declension; and Mw, Nw, and Fw stand for the three weak declensions.
Notes on Paradigm Economy
173
As a matter of fact, very few of the genitive singular or nominative plural
markers in (14) unambiguously identifies and inflection class (the exceptions
are ur and u in the genitive singular, and Ø and u in the nominative plural).6
In view of this state of affairs, one might think that the same kind of solution
can be suggested for the No Blur Principle as we have seen as a possible
reaction to apparent counter-evidence to the Paradigm Economy Principle
(and this kind of solution is indeed adopted by Carstairs-McCarthy (1994)
for other phenomena): The No Blur Principle only holds for inflection classes
of the same gender, not across genders. Still, this does not yet seem to suffice:
In masculine nominative plural contexts, neither ar nor ir unambiguously
identifies inflection class: The former marker shows up in Ma and Mw, the
latter in Mi and Mu. It is not clear whether this problem can be solved in
a way that is not ad hoc.
I take the specific problem just discussed to be indicative of a more general potential problem that is raised by the No Blur Principle (as well as by
Noyer’s (2005) related Interclass Syncretism Constraint – more precisely,
by the assumptions that ultimately derive the latter constraint): Transparadigmatic syncretism (i.e., instances of syncretism that affect more than
one inflection class) is a recurring pattern of inflectional systems. This pattern has successfully been addressed by standard techniques (going back
to Jakobson (1936) and Bierwisch (1967), among others) involving feature
decomposition and underspecification (which permits a reference by inflection marker specifications to natural classes of inflection classes). Proposals involving decomposed inflection class features and concomitant underspecification of inflection class information include Halle (1992) (for transparadigmatic syncretism in Lativan noun inflection), Oltra Massuet (1999)
(for Catalan verb inflection), Wiese (1999) (for German pronominal inflection), Stump (2001) (for Bulgarian verb inflection), Alexiadou & Müller
(2005) (for Russian, Greek, and German noun inflection), Müller (2005)
(for Icelandic noun inflection), and Trommer (2005) (for Amharic verb inflection); also see several of the papers in the present volume (e.g., Börjesson
(2006), Opitz (2006), Weisser (2006)). In all these approaches, more than
one of the inflection markers competing for a given instantiation of a grammatical category fails to unambiguously identify inflection class, in violation
of the No Blur Principle.
6 And even the latter is not uncontroversially class-specific if ur forms are subanalyzed as a combination of two markers, as suggested in Müller (2005): a marker u that
bears (underspecified, i.e., non-identifying) inflecion class information, and a general nonobliqueness marker r. – Also see Bierkandt (2006) and many other contributions to this
volume on the concept of subanalysis.
174
Gereon Müller
To sum up, I think that the Paradigm Economy Principle and the No
Blur Principle (and the related Interclass Syncretism Constraint) can be
viewed as interesting and plausible proposals that reduce the set of logically
possible inflection classes (based on a given inventory of markers) to a very
small set. However, it seems clear that these constraints constantly face the
danger of being too restrictive. Severe undergeneration problems can only
be avoided by assuming that differences between inflection classes which
are independently derivable (by invoking gender features, phonological features, or semantic features) are somehow irrelevant for the constraints; and
this, I believe, may take away something of the constraints’ initial predictive
power. Furthermore, it has turned out that these constraints are incompatible with the view that paradigms are mere epiphenomena (this holds for
the Paradigm Economy Principle, and to some extent also for the No Blur
Principle), and with the view that trans-paradigmatic syncretism can be accounted for by invoking class feature decomposition and underspecification.
All in all, I would like to conclude that this warrants looking for alternative
ways of bringing about paradigm economy. In the next section, I will argue that a version of paradigm economy follows straightforwardly from the
Syncretism Principle in (1).
3.
Paradigm Economy as a Theorem
3.1. Claim
In what follows, I will basically presuppose an approach along the lines of
Distributed Morphology. However, this is mainly to have a theory in which
to frame the discussion. As far as I can see, the issues to be discussed
below arise in exactly the same way – and can be addressed in exactly
the same way – in alternative morphological theories, such as Minimalist
Morphology (Wunderlich (1996, 1997)) or Paradigm Function Morphology
(Stump (2001)). (Where appropriate, I will therefore also discuss these latter
two approaches.) The central claim I would like to advance here is that (15)
holds.
(15)Inflection Class Economy Theorem:
Given a set of n inflection markers, there can be at most 2n−1 inflection
classes, independently of the number of instantiations of the grammatical category that the markers have to distribute over.
The number of 2n−1 inflection classes encodes the powerset of the inventory
of markers, minus one radically underspecified marker. I will explain below
Notes on Paradigm Economy
175
(in section 3.3) why exactly this number would be relevant. For now, we can
conclude that (15) significantly restricts the number of possible inflection
classes over a given inventory of inflection markers. For instance, assuming
an abstract system with five markers and six instantiations of a grammatical
category (e.g., case), the Inflection Class Economy Theorem states that
there can at most be sixteen (i.e., 25−1 = 24 ) inflection classes, out of the
15.625 (i.e., 56 ) that would otherwise be possible.
The Inflection Class Economy Theorem follows under any morphological
theory that makes the three assumptions in (16), (17), and (18), which I call
‘Syncretism’, ‘Elsewhere’, and ‘Uniqueness’. I discuss and try to motivate
them in turn, beginning with Syncretism.
(16)Syncretism:
The Syncretism Principle holds: For each marker, there is a unique
specification of morpho-syntactic features.
The Syncretism Principle underlies much recent (and, based on the Jakobsonian tradition, some not so recent) work in inflectional morphology; it
provides simple and elegant analyses, and it has been empirically confirmed
for a variety of inflectional systems in the world’s languages. Assuming Syncretism to be valid is the starting point of the present paper.
The second assumption is that there is always one radically underspecified (elsewhere) inflection marker in any given morphological domain.
(17)Elsewhere:
There is always one elsewhere marker that is radically underspecified
with respect to inflection class (and more generally). Other markers
may be underspecified to an arbitrary degree (including not at all).
The concept of underspecification as a means to account for syncretism
is employed in most recent theories of inflectional morphology – in Distributed Morphology, Minimalist Morphology, Paradigm Function Morphology, etc.7 Similarly, the assumption that there is always one radically un-
7 In Distributed Morphology (see, e.g., Halle & Marantz (1993), Halle (1997)), functional heads in syntax provide fully specified contexts for insertion of vocabulary items;
and whereas the former are characterized by fully specified morpho-syntactic features (ignoring impoverishment), the vocabulary items can be (and often are) underspecified with
respect to these features; a Subset Principle ensures that a vocabulary item can only be
inserted if it does not bear features which contradict those in the functional morpheme
in syntax. Similarly, underspecification is considered to be one of the central assumptions of Distributed Morphology (see Wunderlich (1996, 2004)). Even though Minimalist
Morphology differes from Distributed Morphology in being an “incremental” approach,
176
Gereon Müller
derspecified elsewhere marker in inflectional systems is quite common, and
well-motivated empirically because it can account for ‘discontinuous’ occurrences of markers in paradigms (where natural classes captured by nonradical underspecification is unlikely to be involved). Furthermore, and
perhaps most importantly, the existence of an elsewhere marker ensures
that there are (usually) no paradigmatic gaps in inflectional systems. As
soon as underspecification is adopted, and the traditional idea is abandoned that all inflection markers are characterized by fully specified morphosyntactic specifications, the situation may arise that there is a fully specified context for which there is no marker that fits into it (such that a
subset/compatibility/extension relation exists; see the last footnote). Such
a situation is avoided in principle if there is always a marker that fits anywhere.8
The third and final assumption that must be made to derive the Inflection Class Economy Theorem is Uniqueness. Every theory of morphology that employs underspecification9 must somehow ensure that only one
where the inflection marker contributes features to the whole word that would otherwise
not be present (see Stump (2001) for the terminology), Wunderlich manages to integrate
underspecification of inflection markers into the system, and in doing so invokes a Compatibility requirement that has effects which are similar to those of the Subset Principle.
Finally, in Paradigm Function Morphology (see Stump (2001)), inflection markers are
added to stems by morphological realization rules, which take the abstract form of (i).
(i) RRn,τ,C (<X,σ>) = <Y′ ,σ>
Here, τ is the set of morpho-syntactic features associated with the inflection marker
(the inflection marker emerges as the difference between the stem X and the inflected
word Y′ ); τ can be underspecified. In contrast, σ is the set of morpho-syntactic features
that the fully inflected word form bears (the analogue to the insertion contexts provided by functional morphemes in Distributed Morphology). Importantly, a constraint
on rule/argument coherence ensures that σ is an extension of τ ; this is comparable to
the subset and compatibility requirements of Distributed Morphology and Minimalist
Morphology, respectively.
8 The assumption that there is always one radically underspecified marker that fits
into any context is often not made explicit, and then only emerges as a by-product of the
feature specifications for inflection markers that is proposed in an analysis. See, however,
Stump (2001), who adopts an Identity Function Default rule to this effect.
9 Or, in fact, any other means that triggers a competition of markers, like non-faithful
feature realization by feature-changing impoverishment in Distributed Morphology (see
(Noyer (1998)) or by optimal violation of faithfulness constraints in recent versions of
Minimalist Morphology (see Wunderlich (2004)); or rules of referral in Paradigm Function Morphology (Stump (2001)) and related approaches like Network Morphology (see
Corbett & Fraser (1993), Baerman et al. (2005)).
Notes on Paradigm Economy
177
marker can be chosen for any given instantiation of a grammatical category (i.e., morpho-syntactic context). The Uniqueness requirement is often
understood in terms of specificity, as in (18).
(18)Uniqueness:
Competition of underspecified markers is resolved by choosing the most
specific marker: For all (competing) markers α, β, either α is more
specific than β, or β is more specific than α.
A Specificity constraint along these lines is adopted in Distributed Morphology (typically as part of the definition of the Subset Principle, see Halle
(1997)), in Minimalist Morphology (see Wunderlich (1996, 1997, 2004)),
and in Paradigm Function Morphology (Stump (2001) calls the relevant
constraint Panini’s Principle). Note that it does not matter how specificity
is to be understood exactly – e.g., whether it is determined by simply counting the number of features that characterize an inflection marker (so that
one marker can be more specific than another one even though they do not
stand in a a subset/extension relation); or by invoking subset/superset relations (so that one marker cannot be more specific than another marker if
the two do not compete; see, e.g., Stump’s (2001) notion of a “narrower”
rule); or by a hierarchy of features (or feature classes).10
Taken together, the three assumptions needed to derive the Paradigm
Economy Theorem amount to stating that (i) syncretism is systematic in the
sense that only one specification of morpho-syntactic features is associated
with any given inflection marker (with the qualifications made in (1)); (ii)
for any given fully specified context, there is always one inflection marker
that fits; and (iii) for any given fully specified context, there is never more
than one inflection marker that fits. Thus, Elsewhere and Uniqueness emerge
as two sides of the same coin.11
With the Inflection Class Economy Theorem and the assumptions needed
to derive it in place, there are two issues that need to be addressed. First,
how does the Inflection Class Economy Theorem constrain inflectional systems (and how does it differ in this respect from the constraints discussed
in section 2)? And second, how does the Inflection Class Economy Theorem
10 In
the same way, extrinsic ordering (as, e.g., in Bierwisch (1967)) could be adopted to
determine the competition winner; or constraint ranking (Wunderlich (2004)); or other
mechanisms, like the expanded mode property of certain realization rules that Stump
(2001) argues for on the basis of, i.a., Georgian prefixal argument encoding morphology.
11 Assumptions (ii) and (iii) correspond to the principles of ‘Completeness’ and ‘Uniqueness’ in Wunderlich (1996, 99).
178
Gereon Müller
follow as a theorem from Syncretism, Elsewhere, and Uniqueness? I address
the former question first.
3.2. Illustration
The basic question underlying all work on paradigm economy comes in two
versions, viz., (19-a) and (19-b):
(19)a.
b.
Given an inventory of markers for a certain domain (e.g., noun
inflection), how many inflection classes can there be?
Given an inventory of markers with associated features encoding a
grammatical category (e.g., case) for a certain domain (e.g., noun
inflection), how many inflection classes can there be?
Carstairs (1987) only tries to answer (19-b); but I find (19-a) the arguably
more interesting question: It does not presuppose that the specification of a
marker for a grammatical category (e.g., with respect to case and/or number) is somehow privileged, i.e., more basic than its inflection class features.
Thus, from the perspective of a child acquiring an inflectional system, (19-b)
would seem to presuppose that there is a stage in the acquisition process
where the child has learned a set of markers together with their syntactically relevant features (such as case and number), and then needs to decide
how all these markers can be assembled into inflection classes. In contrast,
(19-a) suggests that the child is confronted with a set of markers and faces
the task of assigning these markers (typically underspecified) specifications,
including syntactically relevant features and inflection class features that are
only important for morphology. In what follows, I assume that it is question
(19-a) that should be addressed by a theory of paradigm economy.
With this in mind, let me introduce a few abstract examples to illustrate
the workings of the Inflection Class Economy Theorem. In a system without
any restrictions on the number and form of inflection classes, we can observe
the following generalization: If, in a given domain (e.g., noun inflection),
there are n markers for m instantiations of a grammatical category (e.g.,
case), the markers can be grouped into nm distinct inflection classes (i.e.,
the set of m-tuples over an input set with n members). Thus, suppose that
we have a system of noun inflection with three markers and four cases. In
an unconstrained system, there should then be eighty-one logically possible
inflection classes. This is shown in Table 1.12
12 The generation of the abstract lists in this section and in the following section (as
well as of many other, often much more complex lists that were used in developing
Notes on Paradigm Economy
179
Table 1: 3 markers, 4 cases: 81 (= 34 ) possible inflection classes
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
a
a
a
b
b
b
c
c
c
a
a
a
b
b
b
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
c
c
c
c
c
c
c
c
c
a
a
a
c
c
c
a
a
a
b
b
b
c
c
c
a
a
a
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
a
a
a
a
a
a
b
b
b
b
b
b
b
b
b
b
b
b
c
c
c
a
a
a
b
b
b
c
c
c
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
b
b
b
b
b
b
b
b
b
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
a
a
a
a
a
a
a
a
a
b
b
b
c
c
c
a
a
a
b
b
b
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
a
a
a
b
b
b
b
b
b
b
b
b
c
c
c
c
c
c
a
a
a
b
b
b
c
c
c
a
a
a
a
b
c
a
b
c
a
b
c
a
b
c
a
b
c
c
c
c
c
c
c
c
c
c
c
c
c
b
b
b
c
c
c
a
b
c
a
b
c
Here, a, b, and c stand for the three markers; and all four-letter rows
(4-tuples separated by either a vertical line or a line break) correspond to
one inflection class, with the first marker in a row being used for the first
instantiation of case (e.g., nominative), the second one for the second instantiation of case (e.g., accusative), the third one for the third instantiation
of case (e.g., dative), and the fourth one for the fourth instantiation of case
(e.g., genitive). It seems extremely unlikely that a language can be found in
which eighty-one inflection classes have been generated on the basis of three
markers and four instantiations of a grammatical category. The Paradigm
Economy Principle narrows down the class of possible inflection classes from
eighty-one to three. Under present assumptions (where paradigm economy
considerations do not take into account the markers’ grammatical category
specification), the worst case scenario for the Paradigm Economy Principle
is that all three markers can be allomorphs for a single case specification
(e.g., a, b, and c can all be accusative markers); still, there can then only
be three distinct inflection classes.
Next, the No Blur Principle permits maximally nine inflection classes
out of the eighty-one classes in Table 1. In the worst case scenario, there
is one default marker (say, a). One class consists only of default markers
(aaaa), and all the other inflection classes differ from this class by replacing
one of the a’s with either b or c (baaa, abaa, aaba, aaab, caaa, acaa, aaca,
the present analysis) was greatly facilitated by one of Chris Potts’ contributions to the
comp4ling toolbox, available on the website of the University of Massachusetts linguistics
department: http://web.linguist.umass.edu/∼comp4ling/.
180
Gereon Müller
aaac), so that all classes respect the No Blur Principle.13 Adding another
class with more than one b, or more than one c, or a – perhaps minimal –
combination of b’s and c’s (cf. bbaa, or aacc, or abca, etc.) will invariably lead
to a violation of the No Blur Principle because either b or c (or both) will
cease to be inflection-class specific. In general, the No Blur Principle predicts
that there can at most be ((n-1)×m)+1 inflection classes, for n markers and
m instantiations of a grammatical category: Every marker except for one –
the default marker, hence “–1” – can appear for a given instantiation of a
grammatical category only in one inflection class; and “+1” captures a class
consisting exclusively of default markers.
Turning finally to the Inflection Class Economy Theorem, we expect that
at most four (i.e., 23−1 ) classes can exist in the example in Table 1, out of
the eighty-one classes that are a priori possible. The predictions made by
the Paradigm Economy Principle, the No Blur Principle, and the Inflection
Class Economy Theorem for the system underlying Table 1 are summarized
in (20).
(20)a.
b.
c.
Paradigm Economy Principle, worst case scenario:
3 inflection classes: the size of the inventory
No Blur Principle, worst case scenario:
9 inflection classes: ((3-1)×4)+1
Inflection Class Economy Theorem, worst case scenario:
4 inflection classes: 23−1
Consider now a second abstract example. This time, there are five inflection
markers, and four cases (or other instantiations of a grammatical category).
As shown in Table 2, without constraints there could be 125 different inflection classes.
In the worst case scenario under the Paradigm Economy Principle, there
could be five different inflection classes. Under No Blur, there could in prin-
13 To be sure, such a situation would be unexpected if the Syncretism Principle is
adopted since it would not at all be clear how a marker like, e.g., b here could be given a
unique feature specification. However, No Blur’s workings as such do not rely on something like the Syncretism Principle – in fact, as noted above, No Blur is incompatible
with (non-radical) underspecification accounts of trans-paradigmatic syncretism. – Also
note that assuming default markers that are specific with respect to instantiations of a
grammatical category (such that, e.g., a is the default marker for the first instantiation,
b for the second, c for the third, and perhaps again a for the fourth) instead of an extremely general default marker a, as assumed in the main text, does not change things:
This would be compatible with No Blur, but it could not increase the number of possible
inflection classes. In the case at hand, the maximal set of inflection classes would include
abca, bbca, cbca, aaca, acca, abaa, abba, abcb, abcc.
181
Notes on Paradigm Economy
Table 2: 5 markers, 3 cases: 125 (= 53 ) possible inflection classes
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
c
c
c
c
c
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
d
d
d
d
d
e
e
e
e
e
a
a
a
a
a
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
c
c
c
c
c
d
d
d
d
d
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
b
b
b
b
b
c
c
c
c
c
c
c
c
c
c
e
e
e
e
e
a
a
a
a
a
b
b
b
b
b
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
c
d
d
d
d
d
e
e
e
e
e
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
d
d
d
d
d
d
d
d
d
d
d
d
d
d
d
a
a
a
a
a
b
b
b
b
b
c
c
c
c
c
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
d
d
d
d
d
d
d
d
d
d
e
e
e
e
e
d
d
d
d
d
e
e
e
e
e
a
a
a
a
a
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e
b
b
b
b
b
c
c
c
c
c
d
d
d
d
d
a
b
c
d
e
a
b
c
d
e
a
b
c
d
e
e
e
e
e
e
e
e
e
e
e
a
b
c
d
e
ciple be thirteen inflection classes (e.g., assuming a as a default marker,
aaa, baa, aba, aab, caa, aca, aac, daa, ada, aad, eaa, aea, aae). Given the
Inflection Class Economy Theorem, there is a maximum of sixteen inflection
classes (viz., 25−1 ). Again, this is summarized schematically below:
(21)a.
b.
c.
Paradigm Economy Principle, worst case scenario:
5 inflection classes: the size of the inventory
No Blur Principle, worst case scenario:
13 inflection classes: ((5-1)×3)+1
Inflection Class Economy Theorem, worst case scenario:
16 inflection classes: 25−1
Let me finally bring up a third, slightly more complex example. Suppose
that there are five inflection markers that can be distributed over four instantiations of a grammatical category (again, let us say, four cases). Now
there could in principle be 625 distinct inflection classes on the basis of this
inventory, clearly a highly unlikely situation. Consider the list in Table 3.
The Paradigm Economy Principle drastically reduces the number of possible
inflection classes. Since there are still only five markers, five is the maximal
number of allomorphs for a cell, and consequently there can be no more than
five inflection classes. Next, the No Blur Principle predicts that a maximum
of seventeen inflection classes should be possible (e.g., aaaa, baaa, abaa, aaba,
aaab, caaa, acaa, aaca, aaac, daaa, adaa, aada, aaad, eaaa, aeaa, aaea, aaae).
Finally, the Inflection Class Economy Theorem states that there can still
only be at most sixteen inflection classes for this inventory of markers and
for this number of instantiations of a grammatical category. Thus, whereas
the number of instantiations of a grammatical category does matter for
182
Gereon Müller
Table 3: 5 markers, 4 cases: 625 (= 54 ) possible inflection classes
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
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e
Notes on Paradigm Economy
183
the No Blur Principle, it turns out to be irrelevant for the Inflection Class
Economy Theorem (as well as for the Paradigm Economy Principle): If,
say, five markers are distributed over eight instantiations of a grammatical
category, No Blur predicts a maximum of thirty-three classes under a worst
case scenario, whereas the Inflection Class Economy Theorem still stays
at sixteen (and the Paradigm Economy Principle at five). The different
predictions made for Table 3 are given in (22).
(22)a.
b.
c.
Paradigm Economy Principle, worst case scenario:
5 inflection classes: the size of the inventory
No Blur Principle, worst case scenario:
17 inflection classes: ((5-1)×4)+1
Inflection Class Economy Theorem, worst case scenario:
16 inflection classes: 25−1
Concluding so far, the Inflection Class Economy Theorem restricts possible inflection classes in a way that is roughly comparable to the Paradigm
Economy and No Blur Principles. It now remains to be shown that it can
indeed by derived from the assumptions concerning Syncretism, Elsewhere,
and Uniqueness laid out in the previous section.
3.3. Deriving the Inflection Class Economy Theorem
Recall again what the three main assumptions amount to: Syncretism says
that (exceptions apart) only one morpho-syntactic feature specification is
associated with each marker of the inventory for a given morphological domain; Elsewhere states that there is always one marker that in principle fits
into every context of fully specified morpho-syntactic features (which systematically excludes true paradigmatic gaps); and Uniqueness demands that
eventually there is always only one marker that can in fact be used for any
fully specified context of morpho-syntactic features (which systematically
excludes cases of optionality in exponence).
As a first, and crucial, step towards deriving the Inflection Class Economy Theorem from these three assumptions, note that the following holds:
Since each inflection marker M can only be associated with one specification of morpho-syntactic features (because of Syncretism), it follows that
for each inflection marker M and for each inflection class I, it must be the
case that M is either compatible with I or incompatible with I.14 A marker
14 This
holds independently of whether we assume that the morpho-syntactic features
184
Gereon Müller
is compatible with an inflection class I if it bears no inflection class feature, if it bears fully specified inflection class information that completely
characterizes I, or (assuming inflection class decomposition and underspecification; see the references on page 173) if it is characterized by a set of
underspecified inflection class features that is a subset of (or, in Paradigm
Function Morphology, extendable to) the fully specified set of features that
characterize the inflection class. We can say that M is activated for I if it is
compatible with it; and deactivated for I if it is incompatible with it.15
Next, Uniqueness ensures that each inflection class can be defined in
terms of the markers that are active in it: For all competing markers α and
β, it is fixed once and for all by the markers’ feature specifications (and
independently of inflection classes) that either β is more specific than α, or
α is more specific than β. Hence, if the same set of markers is activated for
two inflection classes I1 and I2 , I1 must be identical to I2 . Conversely, since
every marker is either activated or deactivated for any given inflection class,
it also follows that if the same set of markers is deactivated for two inflection
classes I1 and I2 , I1 and I2 must be the same inflection class (because the
same set of markers is then activated for I1 and I2 , because a marker /x/
can only have one specification [ξ], and because specificity relations among
competing markers are fixed).
In order to determine the maximal number of inflection classes on the
basis of a given inventory of markers,16 it now suffices to successively deactivate all possible marker combinations. Thus, starting with the full inventory
of markers, we can proceed by successively deactivating all combinations of
associated with an inflection marker are encoded as a context specification “↔ [ξ]”, with
the marker viewed as a vocabulary item /x/, as in Distributed Morphology; as a lexical
entry of an affix, as in Minimalist Morphology; or as the τ -part of a realization rule of
exponence in Paradigm Function Morphology (see footnote 7). However, as noted above,
for the sake of concreteness I adopt a Distributed Morphology notation here.
15 However, if a marker is activated for an inflection class I, this does not imply that it
will actually be used by I – there may well be more specific markers that block it.
16 The notion of “marker” is to be understood in a somewhat more abstract way that ignores allomorphic variation which is phonologically or morpho-phonologically conditioned
(and not morphologically, as with variation determined by inflection class membership).
For instance, Halle (1994) argues that the marker realizations ov and ej for genitive
plural in Russian are allomorphs whose choice is morpho-phonologically determined; on
this view, there is but a single marker /ov/, accompanied a single underspecified set of
morpho-syntactic features (perhaps involving underspecified inflection class features, as
suggested in Alexiadou & Müller (2005) in order to account for fact that this marker
exhibits trans-paradigmatic syncretism).
Notes on Paradigm Economy
185
markers, which yields class after class. Thus, all markers of the inventory are
compatible with class I1 ; all except for marker a are compatible with class
I2 ; all except for markers a, b are compatible with class I3 ; and so forth.
However, by assumption (Elsewhere), one marker always is the elsewhere
(default) marker: It is compatible with all inflection classes because it is radically underspecified; and therefore it cannot be deactivated by definition.
Consequently, all possible marker deactivation combinations are provided
by the powerset of the set of all the markers of the inventory minus the elsewhere marker: 2n−1 , for n markers. Thus, given a set of n inflection markers,
there can be at most 2n−1 marker deactivation combinations. Since marker
deactivation combinations fully determine possible inflection classes, it now
follows that given a set of n inflection markers, there can be at most 2 n−1
inflection classes.17
This reasoning is entirely independent of the number of instantiations of
the grammatical category (e.g., the number of cases) that a set of markers
needs to distribute over. In contrast to what is the case under the No Blur
Principle, an increase in instantiations of a grammatical category does not
induce an increase in possible inflection classes over a given inventory of
markers; this information is simply irrelevant for determining the maximal
number of inflection classes. Hence, we end up deriving the statement that
given a set of n inflection markers, there can be at most 2n−1 marker deactivation combinations, independently of the number of instantiations of the
grammatical category that the markers have to distribute over. This is the
Inflection Class Economy Theorem.
It may be useful to illustrate this theorem on the basis of a few simple,
abstract examples.
3.4. Examples
3.4.1.
A First Example
Consider again Table 1. In order to illustrate the possible marker deactivation patterns, the case categories are now called 1, 2, 3, and 4. The marker
deactivation combinations are added in (23-c). Given an inventory of three
markers, there are 23−1 = 4 deactivation combinations.
17 Elsewhere thus turns out to be the least important of the three assumptions needed
to derive the Inflection Class Economy Theorem: If it did not hold, we could still derive
that there cannot be more than 2n inflection classes if there are n markers.
186
Gereon Müller
(23)Example 1 revisited:
a. 3 markers: {a, b, c}
b. 4 cases: 1, 2, 3, 4
c. Deactivation combinations: { {b, c}, {b}, {c}, {} }
As a consequence, of the eighty-one inflection classes that would logically
be possible under an unconstrained combination of inflection markers, only
four remain, given Syncretism, Underspecification, and Uniqueness (i.e., the
Inflection Class Economy Theorem). This result holds under any specificityinduced order of the markers, and under any assignment of case features
to markers. Let me go through a couple of possible assignments of case
features to the three markers of the inventory. Suppose, for instance, that
the three markers /a/, /b/, /c/ are characterized by case features as shown
in (24-a):18 /a/ is the elsewhere marker; /b/ is an underspecified marker
that fits into contexts with either a case 1 or a case 2 specification; and /c/
is an underspecified marker that fits for all case feature specifications that
are not 1.19 Given Uniqueness, the markers must be ordered with respect
to specificity. Under simple notions of specificity, /b/ will emerge as more
specific than /c/ (but this could just as well be the other way round), and
/c/ as more specific than the elsewhere marker /a/; see (24-b). Under these
assumptions, the four possible marker deactivation combinations lead to the
four inflection classes shown in (24-c). If /b/ and /c/ are deactivated, an
inflection class aaaa results; if only /b/ is deactivated, c’s show up wherever
they fit, and where they do not fit, a is used: accc; if only /c/ is deactivated,
18 Again,
the notation here follows Distributed Morphology; but the same information
could just as well be rendered in the notation of Minimalist Morphology, or via realiziation
rules in Paradigm Function Morphology.
19 Assuming decomposition, a specification like [12] can be captured by invoking a
decomposed case feature that is shared by cases 1 and 2 (but not by cases 3 and 4).
Things are slightly more involved with specifications like [234]. Incidentally, configurations
of this type, where one would not want to assume that a 3-out-of-4 syncretism represents
the elsewhere case, seem to occur regularly; compare, e.g., the weak masculine singular
declension in German in (9), or the weak feminine and masculine singular declensions in
Icelandic in (14) (see Alexiadou & Müller (2005) on German, Müller (2005) on Icelandic).
This syncretism can be derived by assuming a third binary case feature, in addition to
the two binary case features that are minimally required to derive four cases via crossclassification; alternatively, negation could be employed (¬1). – More generally, note
that, for present purposes, there is no need whatsoever to impose any restriction on
how natural classes can be captured by feature decomposition. That is, even completely
unrestricted (and therefore perhaps linguistically implausible) systems of decomposition
that introduce an enormous number of abstract, decomposed features in order to classify
markers as belonging to a natural class are fully compatible with the present reasoning.
Notes on Paradigm Economy
187
we derive the class bbaa; and if no marker is deactivated, a does not show
up, and the pattern is bbcc (given that /b/ is more specific than /c/). This
exhausts the possibilities. There is no way that an additional inflection class
could arise on the basis of the marker specifications and assumptions about
specificity in (24-ab).
(24)A possible assignment of case specifications to markers:
a. Markers:
(i) /a/ ↔ [ ]
(ii) /b/ ↔ [12]
(iii) /c/ ↔ [234]
b. Specificity:
/b/ > /c/ > /a/
c. Deactivation combinations and inflection classes:
{b, c} → aaaa
{b}
→ accc
{c}
→ bbaa
{}
→ bbcc
Given these three markers, other case specifications or other specificity relations will produce other possible inflection classes, but not more inflection
classes. This is shown in (25), where the three markers have specifications
that differ from those in (24). The set of classes in (25-c) is the absolute
maximum of classes permitted under Syncretism, Elsewhere, and Uniqueness (but there can be fewer classes – e.g., if /b/ and /c/ are never simultaneously deactivated, aaaa will not be an inflection class in the language).
(25)Another possible assignment of case specifications to markers:
a. Markers:
(i) /a/ ↔ [ ]
(ii) /b/ ↔ [234]
(iii) /c/ ↔ [4]
b. Specificity:
/c/ > /b/ > /a/
c. Deactivation combinations and inflection classes:
{b, c} → aaaa
{b}
→ aaac
{c}
→ abbb
{}
→ abbc
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Gereon Müller
3.4.2.
A second example
As a second illustration, consider again the more complex example in table 3,
with five markers distributed over four cases; cf. (26). There are 625 possible
inflection classes in an unconstrained system; but there are only sixteen
(25−1 ) deactivation combinations. Four possible assignments of markers to
cases are listed in (27)–(30).
(26)Example 3 revisited:
a. 5 markers: {a, b, c, d, e}
b. 4 cases: 1, 2, 3, 4,
(27)A possible choice:
(28)Another possible choice:
a. Markers:
a. Markers:
(i) /a/ ↔ [ ]
(i) /a/ ↔ [ ]
(ii) /b/ ↔ [23]
(ii) /b/ ↔ [ ]
(iii) /c/ ↔ [14]
(iii) /c/ ↔ [1]
(iv) /d/ ↔ [3]
(iv) /d/ ↔ [2]
(v) /e/ ↔ [34]
(v) /e/ ↔ [34]
b. Specificity:
b. Specificity:
/d/ > /e/ > /c/ > /b/ > /a/
/c/ > /d/ > /e/ > /b/ > /a/
c. Deactivation combinations
c. Deactivation combinations
& inflection classes:
& inflection classes:
{b, c, d, e} → aaaa
{b, c, d, e} → aaaa
{b, c, d}
→ aaee
{b, c, d}
→ aaee
{b, c, e}
→ aada
{b, c, e}
→ adaa
{b, c}
→ aade
{b, c}
→ adee
{b, d, e}
→ caac
{b, d, e}
→ caaa
{b, d}
→ caee
{b, d}
→ caee
{b, e}
→ cadc
{b, e}
→ cdaa
{b}
→ cade
{b}
→ cdee
{c, d, e}
→ abba
{c, d, e}
→ bbbb
{c, d}
→ abee
{c, d}
→ bbee
{c, e}
→ abda
{c, e}
→ bdbb
{c}
→ abde
{c}
→ bdee
{d, e}
→ cbbc
{d, e}
→ cbbb
{d}
→ cbee
{d}
→ cbee
{e}
→ cbdc
{e}
→ cdbb
{}
→ cbde
{}
→ cdee
In (27), /a/ is again the elsewhere marker, /b/, /c/, and /e/ are underspecified markers that refer to natural classes of cases, and /d/ is a fully
Notes on Paradigm Economy
189
specified case marker. There can be at most sixteen inflection classes, based
on the sixteen possible marker deactivation combinations.
In (28), there are two maximally specific case markers (/c/, /d/), there
is one underspecified case marker (/e/), and one marker that is not specified
for case at all (/b/; however, given Uniqueness, this marker must differ
from /a/ in some way, e.g., with respect to inflection class information, so
that a specificity relation is established between the two). This time, there
are only fifteen instaed of sixteen potential inflection classes. The reason
is that deactivating /b/ and not deactiving anything produce the same
inflection class (to indicate this, one of the two classes is crossed out).
(29)A third possible choice:
(30)A fourth possible choice:
a. Markers:
a. Markers:
(i) /a/ ↔ [ ]
(i) /a/ ↔ [ ]
(ii) /b/ ↔ [234]
(ii) /b/ ↔ [1]
(iii) /c/ ↔ [134]
(iii) /c/ ↔ [2]
(iv) /d/ ↔ [123]
(iv) /d/ ↔ [3]
(v) /e/ ↔ [123]
(v) /e/ ↔ [4]
b. Specificity:
b. Specificity:
/d/ > /e/ > /c/ > /b/ > /a/
/e/ > /d/ > /c/ > /b/ > /a/
c. Deactivation combinations
c. Deactivation combinations
& inflection classes:
& inflection classes:
{b, c, d, e} → aaaa
{b, c, d, e} → aaaa
{b, c, d}
→ eeea
{b, c, d}
→ aaae
{b, c, e}
→ ddda
{b, c, e}
→ aada
{b, c}
→ ddda
{b, c}
→ aade
{b, d, e}
→ cacc
{b, d, e}
→ acaa
{b, d}
→ eeec
{b, d}
→ acae
{b, e}
→ dddc
{b, e}
→ acda
{b}
→ dddc
{b}
→ acde
{c, d, e}
→ abbb
{c, d, e}
→ baaa
{c, d}
→ eeeb
{c, d}
→ baae
{c, e}
→ dddb
{c, e}
→ bada
{c}
→ dddb
{c}
→ bade
{d, e}
→ cbcc
{d, e}
→ bcaa
{d}
→ eeec
{d}
→ bcae
{e}
→ dddc
{e}
→ bcda
{}
→ dddc
{}
→ bcde
(29) represents an extreme case: All markers except /a/ are underspecified
with respect to case, and their specifications overlap to a significant degree.
Now it turns out that the sixteen marker deactivation combinations result in
190
Gereon Müller
only ten distinct inflection classes; e.g., the inflection class that is generated
if /b/ and /c/ are deactivated is identical to the inflection class that is
generated if /b/, /c/ and /e/ are deactivated.
Finally, in (30), all markers (except for /a/) bear fully specified case
information. Again, there cannot be more than sixteen inflection classes
going back to the sixteen marker deactivation combinations.20
3.4.3.
Outlook
To end this paper, let me say something about the scope of the result reported here (viz., that there is a maximum of 2n−1 inflection classes for
a system comprising n markers). As already noted in the context of introducing the Syncretism Principle (1), it seems unavoidable to conclude
that there may be minor imperfections in inflectional systems that can be
traced back to historical factors. In particular, these deviations from optimal design show up in the form of isolated markers that cannot be given
unique specifications, resulting in a case of non-systematic homophony. In
such a situation, the set of possible inflection classes is mildly increased; it is
2n−1+x , for x additional marker specifications required by unresolved, accidental homophony. Still, given that accidental homophony is the exception
rather than the rule, the reduction effect brought about by the Inflection
Class Economy Theorem remains considerable.21
On the other hand, it is worth pointing out again that the 2n−1 formula captures worst case scenarios. Overlapping marker specifications re-
20 The issue of what the decomposed case and inflection class features that encode
the deactivation patterns in systems like (27)–(30) would actually look like is strictly
speaking orthogonal to my present concerns. Still, for the case at hand, in the worst case
there would have to be four binary inflection class features [±α], [±β], [±γ] and [±δ]
whose cross-classification yields the sixteen inflection classes (with individual markers
underspecified as, e.g., [+α]); and two abstract grammatical category features (e.g., case
features such as [±governed], [±oblique], as in Bierwisch (1967)) would suffice for all
systems but (29), where either reference to negated specifications would be necessary, or
a third primitive feature would have to be invoked. See footnotes 19, 21.
21 The same reasoning applies to the use of means like disjunction or negation in marker
specifications (see, e.g., Bierwisch (1967), Wunderlich (1996)), but only if contradictory
feature specifications are involved (see the last footnote); and for the use of variables over
feature values in marker specifications (i.e., α notation – see Chomsky (1965), Chomsky
& Halle (1968) for the original concept, Noyer (1992), Harley (1994), Johnston (1996),
and Alexiadou & Müller (2005) on its use in morphology, and Börjesson (2006), Georgi
(2006), Lahne (2006), Opitz (2006) in the present volume).
Notes on Paradigm Economy
191
duce the number of possible inflection classes further. Moreover, for an inflectional system to fully exploit the logical possibilities for developing inflection classes as they arise under the Inflection Class Economy Theorem is
extremely unlikely – typically, far from all marker deactivation combinations
will be employed.
Finally, a remark may be due on the effects of certain further operations
as they have been introduced in morphological theories. Consider, e.g., the
concept of fission that is sometimes assumed in Distributed Morphology
(see Halle & Marantz (1993) and Noyer (1992) for two different implementations), or the related concept of rule blocks in stem-and-paradigm
accounts (see Anderson (1992), Stump (2001)). Both concepts give rise to
instances of subanalysis, in the sense that what may look like a complex
marker at first sight turns out to be best analyzed as a sequence of smaller
markers, each with its own specifications. Concepts like fission and rule
blocks raise many interesting questions;22 but as long as it is understood
that no more than one inflection class can determine a sequence of subanalyzed markers in each case, this is inconsequential for the present approach.
A somewhat more interesting issue is raised by impoverishment operations
in Distributed Morphology, which (in their standard, non-feature changing
form) delete morpho-syntactic features after syntax, but before morphological insertion takes place. As shown by Trommer (1999), impoverishment
of this type can be reanalyzed as insertion of a highly specific null marker;
if vocabulary insertion implies feature discharge, such cases of null marker
insertion will remove features before other exponents have a chance to be
inserted, and thereby mimic post-syntactic deletion. If so, we may conclude
that each impoverishment rule also increases the set of n’s (for which the
powerset is created) by one.
These qualifications aside, we end up with the conclusion that if languages obey the Syncretism Principle, the number of possible inflection
classes that can be generated over a given inventory of markers is drastically reduced.
22 They do so, e.g., with respect to the issue of extended exponence in non-inferential
theories, such as Distributed Morphology or Minimalist Morphology, which would seem
to require invoking contextual features; but see Müller (2006) for an alternative approach.
192
Gereon Müller
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